This is related to a question I asked earlier - How to end a relationship with a client without pissing them off?

What are your obligations when charging by the hour vs charging by project? If you agree to take on a project, give a rough estimate that it might take 10 days for you to work on and charge £X per hour - are you obligated to work for free after those 10 days are up and you have still not managed to complete your project due to unanticipated issues? What if you have delivered the project but bugs are found - should you fix these bugs for free if the 10 days are up or should you charge your client?

Also, for the above project, what should be the result when you start on the project, but after the 10 days for whatever reason you have to give up and tell your client that you cannot do it anymore? I realise that this does nothing to build your reputation and relationship with the client but are you obligated to pay back the money paid to you or do you just deliver the half/nearly completed source code and help them find someone else to complete it?

The reason I am asking the above questions is because I am very new to freelancing and would like to know how to deal with the above situations if they ever crop up. Thanks!